# Colloquia, AY 2010-2011

 September 2, 2010 Departmental Welcome Event – No colloquium Janet Dyson, University of Oxford, UK Existence and asymptotic properties of solutions of a non-local cell-cell adhesion model in N-dimensional space A model for cell-cell adhesion, based on a model originally proposed by N. J. Armstrong, K. J. Painter, and J. A. Sherratt (2006), is studied. The model consists of a non-linear partial differential equation for the cell density in an N-dimensional infinite domain. It has a nonlocal flux term which models the component of cell motion attributable to cells having formed bonds with other cells within its sensing radius. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments. This is joint work with S. A. Gourley, R. Villella-Bressan and G. F. Webb. Contact person: Glenn Webb Faculty meeting – No colloquium Robert Calderbank, Duke University Something Old, Something New Reed Muller codes are old. They were among the very first algebraic error correcting codes to be discovered and analyzed and they find application today as spreading sequences in spread spectrum wireless communication. Compressed sensing is much more modern. The idea of capturing attributes of a signal with very few measurements has wide applicability and we will describe how second order Reed-Muller codes lead to a new deterministic framework. Contact person: Alex Powell Fall Break – No colloquium Adrian Ioana, University of California, Los Angeles Superrigidity for von Neumann algebras From every countable group G or measure preserving action of G on a probability space X, one can construct a von Neumann algebra. A central theme in the theory of von Neumann algebras is understading how much of the group or group action is “remembered” by its von Neumann algebra. In this talk, I will survey recent results which provide the first classes of groups and group actions that can be completely recovered from their von Neumann algebras. Contact person: Dietmar Bisch Marcin Kozik, Jagiellonian University, Krakow, Poland Universal algebra in constraint satisfaction problems (and vice versa) With every relational structure (e.g. a directed graph) one can associate the set of operations preserving the relations of this structure (the edge-relation in case of a digraph). This correspondence is at the core of a connection between computational complexity of non-uniform constraint satisfaction problems and universal algebra. In the talk I will briefly outline the connection, introduce the most significant advances and discuss the impact on both areas. Contact person: Ralph McKenzie Faculty meeting – No colloquium Thanksgiving Break – No colloquium Xingxing Yu, Georgia Institute of Technology Partitions of graphs and hypergraphs Finding good partitions of graphs and hypergraphs is important to many combinatorial problems and has applications to other fields such as VLSI designs. Graph and hypergraph partitions have been been studied extensively by researchers from various fields using various techniques. In this talk I will discuss a number of such problems. I will also present some recent results we obtained using structural and probabilistic methods. Contact person: Mark Ellingham Tim Austin, Brown University Some recent advances in multiple recurrence In 1975 Szemerédi proved the remarkable combinatorial fact that any subset of the integers having positive upper density contains arbitrarily long arithmetic progressions. Although Szemerédi’s proof was purely combinatorial, shortly afterwards Furstenberg gave a new proof of Szemerédi’s Theorem using a conversion to an assertion of multiple recurrence’ for probability-preserving systems, which he then proved using newly-developed machinery in ergodic theory. Furstenberg’s proof gave rise to a new subdiscipline called Ergodic Ramsey Theory’, which went on to provide proofs for several other extremal results in different combinatorial settings. More recent work has provided a much more detailed picture of the structures that underlie these ergodic theoretic analyses, and offered a clearer insight into the connections between this field and purely combinatorial approaches to the same results. In this talk I will describe some of this interplay and sketch how it has led to both new advances within ergodic theory and to a new approach to the multidimensional generalizations of multiple recurrence and Szemerédi’s Theorem. Contact person: Guoliang Yu Stavros Garoufalidis, Georgia Institute of Technology Asymptotics of classical spin networks A classical spin network is a cubic (i.e. 3-regular) graph whose edges are colored by natural numbers. Its evaluation is an integer. When we scale the colors of all edges by the same amount, we get a sequence of integers, whose asymptotics captures important information about the graph. We will discuss (a) the existence of asymptotic expansions, using the theory of G-functions and algebraic geometry, (b) the computation of asymptotic expansions using the Zeilberger-Wilf theory of holonomic sequences, (d) arithmetic invariants of spin networks and (d) their application to combinatorics and low dimensional topology. We will present numerous examples to illustrate the abstract/concrete principles involved. This is joint work with Roland van der Veen. Contact person: Dietmar Bisch Jan Prüss, Martin-Luther-Universität Halle-Wittenberg, Germany Evolution Equations, Maximal Regularity, and Free Boundary Problems In this survey talk I will explain the basic ideas of the theory of abstract evolution equations as well as present applications to partial differential equations. I intend to show how the concept of maximal regularity naturally comes into play for quasilinear parabolic problems. An outline of its impact on the analysis of free boundary problems will be given. Contact person: Gieri Simonett Spring Break – No colloquium Claude LeBrun, Stony Brook University On Four-Dimensional Einstein Manifolds An Einstein metric is by definition a Riemannian metric of constant Ricci curvature. One would like to completely determine which smooth compact n-manifolds admit such metrics. In this talk, I will describe recent progress regarding the 4-dimensional case. These results specifically concern 4-manifolds that also happen to carry either a complex structure or a symplectic structure. Contact person: Ioana Suvaina Noah Snyder, Columbia University Finite Quantum Groups A common theme in modern mathematics is to pretend that an arbitrary ring is the ring of functions on some space, even if the ring is non-commutative or nilpotent and thus not the ring of functions on an honest space. Similarly, it’s fruitful to look at categories which look like the representation theory of a finite group even if they don’t come from an honest group. Such a category is called a fusion category, and fusion categories can be thought of as the representation theory of a “finite quantum group.” I’ll begin by motivating the definition of a fusion category and giving several examples of fusion categories. The bulk of the talk will be spent justifying the study of fusion categories by explaining how they’re related to other subjects in algebra, topology, and operator algebras. Towards the end of the talk I’ll discuss current research in the field, with examples drawn both from my own work and the work of others. Contact person: Dietmar Bisch Dmitri Burago, Pennsylvania State University Boundary rigidity, volume minimality, and minimal surfaces in L∞: a survey A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric). To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can “tap” at some points of the surface of the body and “listen when the sound gets to other points”. The question is if this information is enough to determine what is inside. This problem has been extensively studied from the PDE viewpoint: the distance between boundary points can be interpreted as a “travel time” for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc. In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first class of boundary rigid metrics of non-constant curvature beyond two dimensions. We are now able to extend this result to include metrics close to a hyperbolic one. The approach grew out of another long-term project of studying surface area functionals in normed spaces, which we have been working on for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifolds. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will not get into any technical details of the proofs). Contact person: Mark Sapir John Weymark, Dept of Economics, Vanderbilt University Dominant Strategy Implementation with a Convex Product Space of Valuations Mechanism design theory is concerned with social decision making when information is privately held by n individuals. Individual i‘s private information is described by his type ti. We consider direct mechanisms that assign an outcome and an n-vector of individual payments to each n-tuple of reported individual types. The allocation function assigning outcomes to types is dominant strategy implementable (DISC) if there is a payment function such that nobody ever has an incentive to falsely report his type. When there are a finite number m of outcomes, for a given individual i and type vector of the other individuals, we can equivalently describe i‘s type ti by a vector vti in Rm, where vjti is the value of the jth outcome to i when i is of type ti. This set of valuation types is used to define a directed graph (the valuation graph) whose nodes are the set of possible outcomes. The Rockafellar-Rochet Theorem provides necessary and sufficient conditions for an allocation function to be DISC in terms of the length of cycles with an arbitrary number of directed arcs in the valuation graph. The Saks-Yu Theorem shows that it is sufficient to only consider 2-cycles if the set of possible valuation vectors is convex. We show that some stronger implications follow if the set of valuations is a convex product set. We also identify and exploit some geometric properties of this problem. This is joint work with Katherine Cuff, Sunghoon Hong, Jesse A. Schwartz and Quan Wen. Contact person: Paul Edelman David Fisher, Indiana University Rigidity of Anosov Actions I will give a survey of history and motivation for the study of rigidity of higher rank group actions. I will then focus on some recent work with Kalinin and Spatzier on rigidity of Anosov actions of higher rank abelian groups on tori and nilmanifolds. Contact person: Guoliang Yu Yang Wang, Michigan State University Peano Curves for Fractals In 1890 G. Peano introduced a continuous function that maps the unit interval onto the unit square. It is the first construction of a space-filling curve. In 1891 Hilbert discovered another space-filling curve by employing the technique of self-similarity. Later space-filling curves have been constructed by many authors, all of which have the properties that they are Hölder continuous and are measure preserving. A natural question is whether a Peano curve exists for any connected self-similar set such as the Sierpinski Gasket . We discuss this question in this talk. We also discuss some interesting applications of such Peano curves. Contact person: Akram Aldroubi